the problem is sec^2x-sec^2x/csc^2x
any help would be great and please show work so ill know how to do other problems :)How do you use the fundemntal identity to change something to an equation with only sines and cosines?
ummmmm....How do you use the fundemntal identity to change something to an equation with only sines and cosines?
sec^2(x) - sec^2(x)/csc^2(x) =
sec^2(x) * [1 - 1/csc^2(x)]
substituting the definitions of sec(x) = 1/cos(x), and csc(x) = 1/sin(x):
sec^2(x) * [1 - 1/csc^2(x)] =
1/cos^2(x) * [1 - sin^2(x)]
From the identity sin^2(x) + cos^2(x) = 1 we know that:
[1 - sin^2(x)] = cos^2(x)
Substituting this into our formula:
1/cos^2(x) * [1 - sin^2(x)] =
1/cos^2(x) * cos^2(x) = 1
If you want to go a bit more advanced, for cos(x) = 0 there appears to be a problem, as sec(x) goes infinite. However, if cos(x) = 0 then sin(x) = 1, so csc(x) = 1, and the term [1 - 1/csc^2(x)] equals 0. Thus, sec^2(x) * [1 - 1/csc^2(x)] appears to become infinity * 0 which could be anything. This requires a bit of care. You can look at the limit as cos(x) approaches 0. Since we've shown above that
sec^2(x) - sec^2(x)/csc^2(x) = 1 for all x, at least if cos(x) does not equal 0, then in the limit as it approaches 0, the expression stays equal to 1. Thus the expression is in fact true at the limit, and therefor:
sec^2(x) - sec^2(x)/csc^2(x) = 1 for all x.
